Relationships Between Variables Notes PDF

Name: Physics 11 M. Lam Relationships Between Variables Block: A relationship in physics describes how a change to one variable will affect another. There are two types of relationships: Direct: The two variables do the same thing. If one varia...

Name: Physics 11 M. Lam Relationships Between Variables Block: A relationship in physics describes how a change to one variable will affect another. There are two types of relationships: Direct: The two variables do the same thing. If one variable increases, so does the other and vice versa. Inverse: The two variables do the opposite. If one variable increases, the other decreases. Relationships are usually written as proportionalities and use the symbol ∝ to symbolize the proportion. To make an equation we add a constant (usually k, if unknown) and change the proportionality sign into an equals sign. Note: ∝ means “proportional to” k is the constant of proportionality Directly Proportional (Linear) We say: y is directly proportional to x Proportionality: y ∝ x Equation: y = kx Meaning: If x increases, y increases proportional to x. For example, if x doubles, y doubles; if x is halved, y is halved. Example: What will be the change in the force of friction if the normal force is decreased by a factor of three? Ff = µ FN Force of friction is directly proportional to normal force. If the normal force decreases, so does the force of friction. If FN is decreased by a factor of three, Ff is decreased by a factor of three (×1/3). © 2015–2024 Mark Lam mrlamphysics.com  Inversely Proportional We say: y is inversely proportional to x (or y is directly proportional to 1/x) 1 Proportionality: y∝ x k Equation: y= x Meaning: If x increases, y decreases proportional to x. For example, if x doubles, y is halved; if x decreases by a factor of three (ie. one third of its original value), y is tripled. Example: What will be the change in the acceleration if the mass is increased by a factor of ve? FNET a= m Acceleration is inversely proportional to mass. If the mass is increased (and the net force is kept constant), then the acceleration decreases. If m is increased by a factor of ve, the acceleration is decreased by a factor of ve (×1/5). Direct Square (Quadratic) We say: y is directly proportional to the square of x Proportionality: y ∝ x 2 Equation: y = kx 2 Meaning: If x increases, y increases proportional to x2. For example, if x doubles, y quadruples; if x decreases by a factor of 3; y decreases by a factor of 9. Example: What will be the change in the kinetic energy if velocity is tripled? 1 2 Ek = mv 2 The kinetic energy is directly proportional to the square of velocity. If v is tripled, Ek is increased by a factor of nine (×9). © 2015–2024 Mark Lam mrlamphysics.com fi fi fi  Inverse Square We say: y is inversely proportional to the square of x 1 Proportionality: y∝ x2 k Equation: y = x2 Meaning: If x increases, y decreases proportional to x2. For example, if x doubles, y decreases by a factor of four; if x decreases by a factor of three, y increases by a factor of nine. Example: What will be the change in the gravitational force if the separation distance is increased by a factor of 10? m1m2 Fg = G r2 Gravitational force is inversely proportional to the square of the separation distance. If r is increased by a factor of ten, Fg is decreased by a factor of 100 (×1/100). © 2015–2024 Mark Lam mrlamphysics.com  Relationships between multiple variables If there is a relationship between multiple variables, all of which change, you can look at each variable independently and then nd their combined effect. Example: For an object in uniform motion, what will be the change in the displacement if the velocity is increased by a factor of four but the time is halved? d = vt Displacement is directly proportional to both velocity and time. If v is increased by a factor of four, d is increased by a factor of four (×4) If t is halved, d is halved (×1/2) (×4)(×1/2) = ×2 The displacement will be doubled. Example: What will be the change in the gravitational eld if the mass is doubled and the radius is tripled? M g=G r2 Gravitational eld strength is directly proportional to the mass and inversely proportional to the square of the distance. If m is doubled, g is doubled (×2). If r is tripled, g is decreased by a factor of nine (×1/9). (×2)(×1/9) = ×2/9 The gravitational eld will be 2/9 of its original value. © 2015–2024 Mark Lam mrlamphysics.com fi fi fi fi